Task number: 4005
A set with \( (q + 1)^2 \) points is given. Each point \( u \) is assigned a set of colors \( L (u) \) of size \( q + 1 \). In addition, for any two different points \( u \), \( v \) applies \( | L (u) \cap L (v) | \leq 1 \).
Prove that points can be colored so that each point \( u \) gets a color from \( L (u) \) and different points are colored with different colors.
See http://mks.mff.cuni.cz/archive/archive.php, year 21, assignment 7, series 2.